This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat

What do you find more trustworthy, experts or numbers, personal 'know-how' or 'objective facts'? Can science claim special authority based on the objectivity of its methods? Are our ethical decisions always better when we strive to be impartial and unbiased? Why should we value objectivity, and is it achievable anyway? These are a few of the thought-provoking questions Guy Axtell asks in this comprehensive new text book, employing examples from the natural and social sciences as well as philosophy. This unique introduction surveys the key issues in a clear and concise way, assessing the nature of objectivity and value of the demand to be impartial decision-makers. Moving beyond the fundamentals, Axtell explores contemporary feminist and social epistemological attempts to 'reconstruct' the concept of objectivity, explains the implications of the so-called science wars for philosophy and the analytical method, and the ethical consequences of these debates. Objectivity is an excellent introduction to one of the most exciting areas of study in philosophy and science today. Students and scholars alike will value this balanced guide to a hotly contested, and vitally important, topic.

Crispin Wright offers an original perspective on the place of “realism” in philosophical inquiry. He proposes a radically new framework for discussing the claims of the realists and the anti-realists. This framework rejects the classical “deflationary” conception of truth yet allows both disputants to respect the intuition that judgments, whose status they contest, are at least semantically fitted for truth and may often justifiably be regarded as true. In the course of his argument, Wright offers original critical discussions of many central concerns of philosophers interested in realism, including the “deflationary” conception of truth, internal realist truth, scientific realism and the theoreticity of observation, and the role of moral states of affairs in explanations of moral beliefs.

Originally published in 1986. Wittgenstein, William James, Thomas Kuhn and John Wisdom share an attitude towards problems in the theory of knowledge which is fundamentally in conflict with the empiricist tradition. They encourage the idea that in understanding the central concepts of epistemology – objectivity, certainty and reasoning – people and their practices matter most. This clash between orthodox empiricism and a freshly inspired pragmatism forms the background to the strands of argument in this book. With these philosophers as a guide, it points to new directions by showing how the theory of knowledge can be shaped around our actions without sacrificing reason’s control over our beliefs.

To what extent are the subjects of our thoughts and talk real? This is the question of realism. In this book, Justin Clarke-Doane explores arguments for and against moral realism and mathematical realism, how they interact, and what they can tell us about areas of philosophical interest more generally. He argues that, contrary to widespread belief, our mathematical beliefs have no better claim to being self-evident or provable than our moral beliefs. Nor do our mathematical beliefs have better claim to being empirically justified than our moral beliefs. It is also incorrect that reflection on the genealogy of our moral beliefs establishes a lack of parity between the cases. In general, if one is a moral antirealist on the basis of epistemological considerations, then one ought to be a mathematical antirealist as well. And, yet, Clarke-Doane shows that moral realism and mathematical realism do not stand or fall together — and for a surprising reason. Moral questions, insofar as they are practical, are objective in a sense that mathematical questions are not, and the sense in which they are objective can only be explained by assuming practical anti-realism. One upshot of the discussion is that the concepts of realism and objectivity, which are widely identified, are actually in tension. Another is that the objective questions in the neighborhood of factual areas like logic, modality, grounding, and nature are practical questions too. Practical philosophy should, therefore, take center stage.